The pp conjecture for spaces of orderings of rational conics

First counterexamples are given to a basic question raised in: M. Marshall, Open questions in the theory of spaces of orderings, J. Symbolic Logic 67 (2002), 341352. The paper considers the space of orderings (X,G) of the function field of a real irreducible conic C over the field Q of rational numbers. It is shown that the pp conjecture fails to hold for such a space of orderings when C has no rational points. In this case, it is shown that the pp conjecture ‘almost holds’ in the sense that, if a pp formula holds on each finite subspace of (X,G), then it holds on each proper subspace of (X,G). For pp formulas which are product-free and 1-related, the pp conjecture is known to be true, at least if the stability index is finite (M. Marshall, Local-global properties of positive primitive formulas in the theory of spaces of orderings, to appear). The counterexamples constructed here are the simplest sort of pp formulae which are not product-free and 1-related.